# Computational Social Choice Part III Committee Elections Piotr

- Slides: 77

Computational Social Choice (Part III: Committee Elections) Piotr Faliszewski AGH University Kraków, Poland [email protected] edu. pl Recent Advances in Parametrized Complexity – Tel Aviv 2017

Single-Winner Elections Multiwinner Elections movies on a plane voters’ preferences fundamentally different! voting rule Seek the best, the most widely supported candidates t n e m a i rl pa P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Multiwinner Voting: A New Challenge for Social Choice Theory, Trends in Computational Social Choice 2017 s making a shortlist

Which movies to put on a plane? P. Skowron, P. Faliszewski, J. Lang, Finding a collective set of items: From proportional multirepresentation to group recommendation. Artificial Intelligence, 2016 We have the following profile of utilities: 10 8 0 5 0 0 12 3 7 3 4 3 2 3 8 9 I can only put 2 movies on the entertainment system… which ones to pick? Really? Just 2? What sort of planes did we buy? !?

Which movies to put on a plane? We have the following profile of utilities: How to pick two items? Total utility? 10 8 0 5 19 0 0 12 3 14 7 3 4 3 24 2 3 8 9 20

Which movies to put on a plane? We have the following profile of utilities: 10 8 0 5 How to pick two items? A good movie for everyone? 10+7 0 0 12 3 8 + 3 7 3 4 3 12+8 2 3 8 9 + 3

Which movies to put on a plane? We have the following profile of utilities: 10 8 0 5 0 0 12 3 How to pick two items? A customer watches a movie with probability p and chooses to stop watching with probability 1 -p. 7 3 4 3 0 movies : (1 -p) 1 movie : p(1 -p) 2 movies : p 2(1 -p) 2 3 8 9 We should optimize the expected utility!

How to capture all these settings? OWA operators – generalized averages Sequence of numbers: u = ( u 1, u 2, u 3, …, uk ) sort u’ = (u’ 1, u’ 2, u’ 3, …, u’k) ( 3, 1, 7, 5, 2) ( 7, 5, 3, 2, 1) OWA operator α = (α 1, α 2, …, αk) α 1 u’ 1 + α 2 u’ 2 + α 3 u’ 3 + … + αku’k

How to capture all these settings? OWA operators – generalized averages Sequence of numbers: u = ( u 1, u 2, u 3, …, uk ) sort u’ = (u’ 1, u’ 2, u’ 3, …, u’k) ( 3, 1, 7, 5, 2) Artihmetic average OWA operator α = (α 1, α 2, …, αk) ( 7, 5, 3, 2, 1) ( 1/5, 1/5) α 1 u’ 1 + α 2 u’ 2 + α 3 u’ 3 + … + αku’k = 3. 6

How to capture all these settings? OWA operators – generalized averages Sequence of numbers: u = ( u 1, u 2, u 3, …, uk ) sort u’ = (u’ 1, u’ 2, u’ 3, …, u’k) ( 3, 1, 7, 5, 2) Maximum OWA operator α = (α 1, α 2, …, αk) ( 7, 5, 3, 2, 1) ( 1, 0, 0) α 1 u’ 1 + α 2 u’ 2 + α 3 u’ 3 + … + αku’k = 7

How to capture all these settings? OWA operators – generalized averages Sequence of numbers: u = ( u 1, u 2, u 3, …, uk ) sort u’ = (u’ 1, u’ 2, u’ 3, …, u’k) ( 3, 1, 7, 5, 2) Minimum OWA operator α = (α 1, α 2, …, αk) ( 7, 5, 3, 2, 1) ( 0, 0, 1) α 1 u’ 1 + α 2 u’ 2 + α 3 u’ 3 + … + αku’k = 1

How to capture all these settings? OWA operators – generalized averages Sequence of numbers: u = ( u 1, u 2, u 3, …, uk ) sort u’ = (u’ 1, u’ 2, u’ 3, …, u’k) ( 3, 1, 7, 5, 2) Median OWA operator α = (α 1, α 2, …, αk) ( 7, 5, 3, 2, 1) ( 0, 0, 1, 0, 0) α 1 u’ 1 + α 2 u’ 2 + α 3 u’ 3 + … + αku’k = 3

How to capture all these settings? OWA operators – generalized averages Sequence of numbers: u = ( u 1, u 2, u 3, …, uk ) sort u’ = (u’ 1, u’ 2, u’ 3, …, u’k) ( 3, 1, 7, 5, 2) Our probability p of watching a movie example, p = 0. 1 OWA operator α = (α 1, α 2, …, αk) ( 7, 5, 3, 2, 1) ( 1, p, p 2, p 3, p 4) α 1 u’ 1 + α 2 u’ 2 + α 3 u’ 3 + … + αku’k = 7. 5321

OWA Winner Problem Definition N = set of agents What is A = set of items probl the compl em? exity G of o a o lgorit k = total number of items we can hms? d approxi the matio FPT a l n g orithm pick s? α = OWA operator (size k) Let C be a set of k items ui(C) = α({ui(a) | a in C}) For each agent i we have a utility function: ui : A N Goal Find a set C of k items that maximizes u(C) = Σui(C)

Special utility functions • Approval utilities 0 1 1 1 0 1 0 0 0 1 Essentially the hardest case. Most problems are NP-hard and hard to approximate (in an appropriate sense) • Borda utilities The „easiest” case 0 1 2 3 2 3 0 1 2 1 3 0 1 0 2 3 Still NP-hard, but very good approximations possilbe (PTASes in many cases) └─e. g. , for OWAs with a fixed number of nonzero entries

Special utility functions • Approval utilities 0 1 1 1 0 1 Special OWA families • 1 -Best OWA (1, 0, … , 0) • t-Best OWA (1, … , 1, 0, …, 0) t • k-Best OWA (1, 1, …, 1) • (k-1)-Best OWA (1, …, 1, 0) 1 0 1 0 0 0 1 • Borda utilities • t-Median OWA (0, … 0, 1, 0, … 0) t-1 • k-Median OWA (0, …, 0, 1) (minimum) 0 1 2 3 2 3 0 1 2 1 3 0 1 0 2 3 • Hurwicz OWA (x, 0, …, 0, 1 -x) • geometric OWA (p 0, p 1, p 2, …, pk-1) • nonincreasing OWAs

Special OWA families P-time algorithm NP-complete (even for Borda) PTAS for Borda t/k-approximation ↓ (k-1)/k-approximation (PTAS) • 1 -Best OWA (1, 0, …, 0) • t-Best OWA (1, … , 1, 0, …, 0) t • k-Best OWA (1, 1, …, 1) • (k-1)-Best OWA (1, …, 1, 0) • t-Median OWA (0, … 0, 1, 0, … 0) t-1 • k-Median OWA (0, …, 0, 1) Almost no hope for good approximation PTAS for Borda (1 -1/e)-approximation through submodular functions (minimum) • Hurwicz OWA (x, 0, …, 0, 1 -x) • geometric OWA (p 0, p 1, p 2, …, pk-1) • nonincreasing OWAs

Special OWA families P-time algorithm NP-complete (even for Borda) PTAS for Borda t/k-approximation ↓ (k-1)/k-approximation (PTAS) • 1 -Best OWA (1, 0, …, 0) • t-Best OWA (1, … , 1, 0, …, 0) t • k-Best OWA (1, 1, …, 1) • (k-1)-Best OWA (1, …, 1, 0) • t-Median OWA (0, … 0, 1, 0, … 0) t-1 • k-Median OWA (0, …, 0, 1) Almost no hope for good approximation PTAS for Borda (1 -1/e)-approximation through submodular functions (minimum) • Hurwicz OWA (x, 0, …, 0, 1 -x) • geometric OWA (p 0, p 1, p 2, …, pk-1) • nonincreasing OWAs

k-Best: Polynomial Time Algorithm We simply sum up the scores of all the items and pick the k ones with highest scores 0 1 2 3 2 3 0 1 2 1 3 0 1 0 2 3 5 5 7 7 Models several well known multiwinner voting rules: • k-Borda • Bloc • SNTV

Comparison of Voting Rules 2 D-Domain Experiment: candidates: light gray, voters: dark gray, committee: red k-Borda Bloc k-Borda – sum up Borda scores of the candidates Bloc – k-Approval 1 point to each of top k candidates E. Elkind, P. Faliszewski, JR. Laslier, P. Skowron, A. Slinko, and N. Talmon: What Do Multiwinner Voting Rules Do? An Experiment Over the Two-Dimensional Euclidean Domain, AAAI 2017

Comparison of Voting Rules 2 D-Domain Experiment: candidates: light gray, voters: dark gray, committee: red Problem with Bloc k-Borda – sum up Borda scores of the candidates Bloc – k-Approval 1 point to each of top k candidates

Comparison of Voting Rules 2 D-Domain Experiment: candidates: light gray, voters: dark gray, committee: red Problem with Bloc k-Borda – sum up Borda scores of the candidates Bloc – k-Approval 1 point to each of top k candidates

Comparison of Voting Rules 2 D-Domain Experiment: candidates: light gray, voters: dark gray, committee: red k-Borda Bloc 1 -AV (SNTV) k-Borda – sum up Borda scores of the candidates Bloc – k-Approval SNTV uses Plurality scores 1 point to each of top k candidates 1 point to the top candidate

Comparison of Voting Rules 2 D-Domain Experiment: candidates: light gray, voters: dark gray, committee: red k-Borda Bloc 1 -AV (SNTV) k-Borda – sum up Borda scores of the candidates Bloc – k-Approval SNTV uses Plurality scores 1 point to each of top k candidates 1 point to the top candidate

Special OWA families P-time algorithm NP-complete (even for Borda) PTAS for Borda t/k-approximation ↓ (k-1)/k-approximation (PTAS) • 1 -Best OWA (1, 0, …, 0) • t-Best OWA (1, … , 1, 0, …, 0) t • k-Best OWA (1, 1, …, 1) • (k-1)-Best OWA (1, …, 1, 0) • t-Median OWA (0, … 0, 1, 0, … 0) t-1 • k-Median OWA (0, …, 0, 1) (minimum) • Hurwicz OWA (x, 0, …, 0, 1 -x) • geometric OWA (p 0, p 1, p 2, …, pk-1) • nonincreasing OWAs

(k-1)-Best: Already Hard! A bit of a silly rule, but very similar to k-Best, yet already NP-hard Consider how the rule works: 0 4 1 2 3 =7 2 3 0 1 4 =7 2 1 4 3 0 =5 =5 1 0 2 4 3 Take committee: Total score = 24

(k-1)-Best: Already Hard! Reduction from Vertex. Cover v 1 k = 3 v 5 v 2 v 3 v 4

(k-1)-Best: Already Hard! e 3 e 4 e 6 e 5 v 3 v 5 e 7 v 4 v 1 v 2 v 3 v 4 v 5 e 1 1 1 0 0 0 g 1 0 0 1 1 1 e 2 0 1 0 0 1 g 2 1 0 1 1 0 e 3 0 1 0 g 3 1 0 1 e 4 0 0 0 1 1 g 4 1 1 1 0 0 e 5 0 1 1 0 0 g 5 1 0 0 1 1 e 6 0 0 1 g 6 1 1 0 e 7 0 0 1 1 0 g 7 1 1 0 0 1 s e 2 e tiv k = 3 na v 2 v 1 er Reduction from Vertex. Cover alt Utility functions

(k-1)-Best: Already Hard! e 3 e 4 e 6 e 5 v 3 v 5 e 7 v 4 Utility = kn (n = number of edges) s e 2 e tiv k = 3 na v 2 v 1 er Reduction from Vertex. Cover alt Utility functions v 1 v 2 v 3 v 4 v 5 e 1 1 1 0 0 0 g 1 0 0 1 1 1 e 2 0 1 0 0 1 g 2 1 0 1 1 0 e 3 0 1 0 g 3 1 0 1 e 4 0 0 0 1 1 g 4 1 1 1 0 0 e 5 0 1 1 0 0 g 5 1 0 0 1 1 e 6 0 0 1 g 6 1 1 0 e 7 0 0 1 1 0 g 7 1 1 0 0 1 3 (=k) 3 3 3

(k-1)-Best: Already Hard! e 3 e 4 e 6 e 5 v 3 v 5 e 7 v 4 s e 2 e tiv k = 3 na v 2 v 1 er Reduction from Vertex. Cover alt Utility functions v 1 v 2 v 3 v 4 v 5 e 1 1 1 0 0 0 g 1 0 0 1 1 1 e 2 0 1 0 0 1 g 2 1 0 1 1 0 e 3 0 1 0 g 3 1 0 1 e 4 0 0 0 1 1 g 4 1 1 1 0 0 e 5 0 1 1 0 0 g 5 1 0 0 1 1 e 6 0 0 1 g 6 1 1 0 e 7 0 0 1 1 0 g 7 1 1 0 0 1 3 (=k) 3 3 3

(k-1)-Best: Already Hard! e 3 e 4 e 6 e 5 v 3 v 5 e 7 v 4 Utility ≤ kn-1 (n = number of edges) s e 2 e tiv k = 3 na v 2 v 1 er Reduction from Vertex. Cover alt Utility functions v 1 v 2 v 3 v 4 v 5 e 1 1 1 0 0 0 g 1 0 0 1 1 1 e 2 0 1 0 0 1 g 2 1 0 1 1 0 e 3 0 1 0 g 3 1 0 1 e 4 0 0 0 1 1 g 4 1 1 1 0 0 e 5 0 1 1 0 0 g 5 1 0 0 1 1 e 6 0 0 1 g 6 1 1 0 e 7 0 0 1 1 0 g 7 1 1 0 0 1 2 (=k-1) 3 3 3

(k-1)-Best: Already Hard! e 3 e 4 e 6 e 5 v 3 v 5 e 7 v 4 Stays N utiliti P-complete es f much as well, bu or Borda m t same ore techni through a c idea, thoug al proof (t he h) s e 2 e tiv k = 3 na v 2 v 1 er Reduction from Vertex. Cover alt Utility functions v 1 v 2 v 3 v 4 v 5 e 1 1 1 0 0 0 g 1 0 0 1 1 1 e 2 0 1 0 0 1 g 2 1 0 1 1 0 e 3 0 1 0 g 3 1 0 1 e 4 0 0 0 1 1 g 4 1 1 1 0 0 e 5 0 1 1 0 0 g 5 1 0 0 1 1 e 6 0 0 1 g 6 1 1 0 e 7 0 0 1 1 0 g 7 1 1 0 0 1 2 (=k-1) 3 3 3

(k-1)-Best: Already Hard! e 3 e 4 e 6 e 5 v 3 v 5 e 7 v 4 Hard for parameter committee size? s e 2 e tiv k = 3 na v 2 v 1 er Reduction from Vertex. Cover alt Utility functions v 1 v 2 v 3 v 4 v 5 e 1 1 1 0 0 0 g 1 0 0 1 1 1 e 2 0 1 0 0 1 g 2 1 0 1 1 0 e 3 0 1 0 g 3 1 0 1 e 4 0 0 0 1 1 g 4 1 1 1 0 0 e 5 0 1 1 0 0 g 5 1 0 0 1 1 e 6 0 0 1 g 6 1 1 0 e 7 0 0 1 1 0 g 7 1 1 0 0 1 2 (=k-1) 3 3 3

NP-complete +approximation +FPT Special OWA families • 1 -Best OWA (1, 0, …, 0) • t-Best OWA (1, … , 1, 0, …, 0) t NP-complete (even for Borda) PTAS for Borda t/k-approximation ↓ (k-1)/k-approximation (PTAS) • k-Best OWA (1, 1, …, 1) • (k-1)-Best OWA (1, …, 1, 0) • t-Median OWA (0, … 0, 1, 0, … 0) t-1 • k-Median OWA (0, …, 0, 1) (minimum) • Hurwicz OWA (x, 0, …, 0, 1 -x) • geometric OWA (p 0, p 1, p 2, …, pk-1) • nonincreasing OWAs

1 -Best: Chamberlin—Courant Rule (CC) Consider how the rule works: 0 4 1 2 3 =4 2 3 0 1 4 =4 2 1 4 3 0 =4 =3 1 0 2 4 3 Take committee: Total score = 15

1 -Best: Chamberlin—Courant Rule Consider how the rule works: 0 4 1 2 3 =4 2 3 0 1 4 =4 2 1 4 3 0 =4 =3 1 0 2 4 3 Take committee: Total score = 15

1 -Best: Chamberlin—Courant Rule 4 3 2 1 > > > 0 4 1 2 3 2 3 0 1 4 2 1 4 3 0 1 0 2 4 3 > > > > > [LB 16] T. Lu and C. Boutilier: Budgeted Social Choice: From Consensus to Personalized Decision Making, IJCAI 2011 0

1 -Best: Greedy. CC Approximation Algorithm (1 -1/e) 4 3 2 1 Algorithm: > > for i = 1 to k do: 0 4 1 2 3 pick a candidate c that increases 2 3 0 1 4 the score of the assignment most 2 1 4 3 0 return the computed assignment 1 0 2 4 3 > > > > > [LB 16] T. Lu and C. Boutilier: Budgeted Social Choice: From Consensus to Personalized Decision Making, IJCAI 2011 0

1 -Best: Greedy. CC Approximation Algorithm (1 -1/e) Greedy CC 4 3 2 1 Algorithm: > > for i = 1 to k do: pick a candidate c that increases the score of the assignment most > > > return the computed assignment > > The apporximation ratio follows from the classic result [LB 16] T. Lu and C. Boutilier: Budgeted Social Choice: From Consensus to Personalized regarding greedy optimization for submodular functions. Decision Making, IJCAI 2011 0

Chamberlin—Courant PTAS (Borda Utilities) Rank x 1 2 3 m v 1 v 2 Goal: pick K winners among m candidates, to get the highest utility Initialize: Forget about the whole profile beyond rank x: x = mw(K) / K (w(K) is Lambert’s W function, O(log K)) Loop: Keep picking the candidate that appears in the „available” part of the profile most frequently. vn P. Faliszewski, P. Skowron, A. Slinko: Achieving Fully Proportional Representation: Approximability Results, Artificial Intelligence 2015

Chamberlin—Courant PTAS (Borda Utilities) Rank x 1 2 3 m v 1 v 2 Goal: pick K winners among m candidates, to get the highest utility Initialize: Forget about the whole profile beyond rank x: x = mw(K) / K (w(K) is Lambert’s W function, O(log K)) Loop: Keep picking the candidate that appears in the „available” part of the profile most frequently. vn P. Faliszewski, P. Skowron, A. Slinko: Achieving Fully Proportional Representation: Approximability Results, Artificial Intelligence 2015

Chamberlin—Courant PTAS (Borda Utilities) Rank x 1 2 3 m v 1 v 2 Goal: pick K winners among m candidates, to get the highest utility Initialize: Forget about the whole profile beyond rank x: x = mw(K) / K (w(K) is Lambert’s W function, O(log K)) Loop: Keep picking the candidate that appears in the „available” part of the profile most frequently. vn Guarantee: n(m-1)(1 – 2 w(K)/K) utility P. Faliszewski, P. Skowron, A. Slinko: Achieving Fully Proportional Representation: Approximability Results, Artificial Intelligence 2015

Chamberlin—Courant PTAS (Borda Utilities)

Performance in Practice Greedy CC Algorithm P

CC Greedy 1 -1/e PTAS

Ranging. CC Rank x 1 2 3 m v 1 v 2 vn

CC Greedy. CC Algorithm P Ranging. CC

What About Approval Utilities?

Borda versus Approval Utility for item ranked at a given position Why are Borda utilities easier to deal with than approval utilities? (x, y)-non-finicky utilities: At least fraction x of the items get fraction y of the highest utility Approval Bo rda 1 2 3 4 5 6 7 8 9 Item rank

Approval-Based Chamberlin—Courant Set Cover 3 1 4 6 2 5

Approval-Based Chamberlin—Courant Set Cover 3 1 4 6 2 5 1 -Best (Approval) v 1 : v 2 : v 3 : v 4 : v 5 : v 6 : k = number of sets we can use

Approval-Based Chamberlin—Courant Set Cover 3 1 4 6 2 5 1 -Best (Approval) v 1 : v 2 : v 3 : v 4 : v 5 : v 6 : k = number of sets we can use

Approval-Based Chamberlin—Courant For approval utilities we inherit all hardness results from Set. Cover/Max. Cover: 1) NP-hardness 2) W[2]-hardness (k) 3) (1 -1/e)-approx. upper bound N. Betzler, A. Slinko, J. Uhlmann: On the Computation of Fully Proportional Representation, Journal of Artificial Intelligence Research 2013 1 -Best (Approval) v 1 : v 2 : v 3 : v 4 : v 5 : v 6 :

Approval-Based Chamberlin—Courant The only chance to get better results is through FPT algorithms Parameter: m = #candidates Trivial bruteforce approach. Try all subset of k candidates: mk ≤ mm 1 -Best (Approval) v 1 : v 2 : v 3 : v 4 : v 5 : v 6 :

Approval-Based Chamberlin—Courant The only chance to get better results is through FPT algorithms Parameter: n = #agents More involved (but still simple). 1. Partition agents into at most k+1 groups (k ≤ n) (guess). 2. For all but the last group, find a commonly approved candidate. 3. The last group are agents wihtout good representatitve 1 -Best (Approval) v 1 : v 2 : v 3 : v 4 : v 5 : v 6 :

Approval-Based Chamberlin—Courant Parametrization by committee size? W[2]-hardness … but assume that each voter approves at most p candidates (where p is a constant) Algorithm 1 Pick (2 pk/(1 -β)+k)k candidates that are most frequently approved. Try all size-k committees of these candidates. Output the best one FPT Approximation Scheme! Finds β-approximate solution in FPT time P. Skowron, P. Faliszewski: Chamberlin--Courant Rule with Approval Ballots: Approximating the Max. Cover Problem with Bounded Frequencies in FPT Time, Journal of Artificial Intelligence Research 2017

Approval-Based Chamberlin—Courant Parametrization by committee size? W[2]-hardness … but assume that each voter approves at most p candidates (where p is a constant) … what if we want to minimize the number of voters without representative? Algorithm 1 Pick (2 pk/(1 -β)+k)k candidates that are most frequently approved. Try all size-k committees of these candidates. Output the best one Algorithm 2 Search tree algorithm 1) Choose a voter withour representative (randomly) 2) Branch into ≤p subtrees, one for each candidate he/she approves of.

Approval-Based Chamberlin—Courant v 2 v 3 Algorithm 1 Pick (2 pk/(1 -β)+k)k candidates that are most frequently approved. Try all size-k committees of these candidates. Output the best one Algorithm 2 Search tree algorithm 1) Choose a voter withour representative (randomly) 2) Branch into ≤p subtrees, one for each candidate he/she approves of.

Approval-Based Chamberlin—Courant v 2 v 3 Algorithm 1 Pick (2 pk/(1 -β)+k)k candidates that are most frequently approved. Try all size-k committees of these candidates. Output the best one Algorithm 2 Search tree algorithm 1) Choose a voter withour representative (randomly) 2) Branch into ≤p subtrees, one for each candidate he/she approves of.

Approval-Based Chamberlin—Courant v 3 v 2 Algorithm 1 v 1 Pick (2 pk/(1 -β)+k)k candidates that are most frequently approved. Try all size-k committees of these candidates. Output the best one Algorithm 2 Search tree algorithm 1) Choose a voter without representative (randomly) 2) Branch into ≤p subtrees, one for each candidate he/she approves of.

Approval-Based Chamberlin—Courant v 3 v 2 Algorithm 1 v 1 Pick (2 pk/(1 -β)+k)k candidates that are most frequently approved. Try all size-k committees of these candidates. Why does it work? Intuition: If there is a solution that gives a representative to every voter, then the algorithm will find it For each sequence of voters (irrespective of the order) there are choices of good candidates, and the algorithm will try them Output the best one Algorithm 2 Search tree algorithm 1) Choose a voter without representative (randomly) 2) Branch into ≤p subtrees, one for each candidate he/she approves of.

Approval-Based Chamberlin—Courant v 3 v 2 Algorithm 1 v 1 Pick (2 pk/(1 -β)+k)k candidates that are most frequently approved. Try all size-k committees of these candidates. Why does it work? P – probability of finding β-approximate solution C* – an optimal committee N* – voters who have representatives under C* U* – voters who do not have representatvies under C* X – voters without representatives at some node in the tree. If less than (β-1)/β fraction of X belong to N*, then more than (1/β)X belongs to U*, so (1/β)X ≤ U*, so X ≤ βU* Output the best one Algorithm 2 Search tree algorithm 1) Choose a voter without representative (randomly) 2) Branch into ≤p subtrees, one for each candidate he/she approves of.

Generalized in: P. Skowroni: FPT approximation schemes for Approval-Based Chamberlin—Courant maximizing submodular functions, Information and Computation 2017 v 3 v 2 Algorithm 1 v 1 Pick (2 pk/(1 -β)+k)k candidates that are most frequently approved. Try all size-k committees of these candidates. Why does it work? P – probability of finding β-approximate solution C* – an optimal committee N* – voters who have representatives under C* U* – voters who do not have representatvies under C* X – voters without representatives at some node in the tree. If always more than (β-1)/β fraction of X belong to N*, then P ≥ ((β-1)/β)k So (β/(β-1))k trials for a high probability of success Output the best one Algorithm 2 Search tree algorithm 1) Choose a voter without representative (randomly) 2) Branch into ≤p subtrees, one for each candidate he/she approves of.

NP-complete +approximation +FPT Special OWA families • 1 -Best OWA (1, 0, …, 0) • t-Best OWA (1, … , 1, 0, …, 0) t • k-Best OWA (1, 1, …, 1) • (k-1)-Best OWA (1, …, 1, 0) • t-Median OWA (0, … 0, 1, 0, … 0) t-1 • k-Median OWA (0, …, 0, 1) (minimum) • Hurwicz OWA (x, 0, …, 0, 1 -x) NP-complete +approximation • harmonic OWA (1, 1/2, 1/3, …, 1/k) • nonincreasing OWAs

Special OWA families • 1 -Best OWA (1, 0, …, 0) • t-Best OWA (1, … , 1, 0, …, 0) t • k-Best OWA (1, 1, …, 1) • (k-1)-Best OWA (1, …, 1, 0) • t-Median OWA (0, … 0, 1, 0, … 0) t-1 • k-Median OWA (0, …, 0, 1) (minimum) • Hurwicz OWA (x, 0, …, 0, 1 -x) NP-complete +approximation • harmonic OWA (1, 1/2, 1/3, …, 1/k) • nonincreasing OWAs

Overview for Parametrized Complexity #candidates – not much left for OWA rules (a lot of left for approximation algorithms and other rules!) #voters – established for basic rules (but remain open for others—active study!) Special OWA families • 1 -Best OWA (1, 0, …, 0) • t-Best OWA (1, … , 1, 0, …, 0) t • k-Best OWA (1, 1, …, 1) • (k-1)-Best OWA (1, …, 1, 0) • t-Median OWA (0, … 0, 1, 0, … 0) t-1 #committee size – not much hope for FPT (but try (k-1)-best) • k-Median OWA (0, …, 0, 1) FPT-AS for nonincreasing OWAs, but not-nonincreasing are waiting! • Hurwicz OWA (x, 0, …, 0, 1 -x) (minimum) Go! Go! • harmonic OWA (1, 1/2, 1/3, …, 1/k) • nonincreasing OWAs

k-Borda Chamberlin—Courant OWA PAV E. Elkind, P. Faliszewski, P. Skowron, A. Slinko, Properties of Multiwinner Voting Rules, Social Choice and Welfare, 2017 SNTV Bloc Committee Scoring Rules Approval-Based Counting Rules Committee Elections Piotr Faliszewski, Piotr Skowron, Arkadii M. Slinko, Nimrod Talmon: Multiwinner Analogues of the Plurality Rule: Axiomatic and Algorithmic Perspectives. AAAI 2016 P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 Martin Lackner, Piotr Skowron: Consistent Approval-Based Multi-Winner Rules. ar. Xiv 2017 P. Skowron, P. Faliszewski, A. Slinko, Axiomatic Characterization of Committee Scoring Rules, ar. Xiv 2016

Approximation algorithms Greedy. CC k-Borda Monroe Chamberlin—Courant Greedy. Monroe OWA PAV SNTV Bloc Committee Scoring Rules STV Approval-Based Counting Rules STV V 1: Committee Elections STV: Candidates ranked first quota times (about n/(k+1)+1) get elected. If there are no such candidates, delete one ranked first least frequently V 2: V 3: V 4: V 5: V 6:

Approximation algorithms Greedy. CC k-Borda Monroe Chamberlin—Courant Greedy. Monroe OWA PAV SNTV Bloc Committee Scoring Rules LSE-Maximin STV Approval-Based Counting Rules LSE-Copeland Condorcet. Inspired Rules NED SEO Committee Elections S. Sekar, S. Sikdar, L. Xia, Condorcet Consistent Bundling with Social Choice, AAMAS 2017 H. Aziz, E. Elkind, P. Faliszewski, M. Lackner, P. Skowron, The Condorcet Principle for Multiwinner Elections: From Shortlisting to Proportionality, IJCAI 2017.

Approximation algorithms Greedy. CC k-Borda Monroe Chamberlin—Courant Greedy. Monroe OWA Destructive Swap Bribery: If a rule is NP-hard to compute(*) then sometimes a single swap replaces unbounded numer of candidates in a committee PAV SNTV A. D. Procaccia, J. S. Rosenschein, A. Zohar, Multi-Winner LSE-Maximin Elections: Complexity of Manipulation, Control and Winner. Bloc NED Approval-Based LSE-Copeland Determination, IJCAI 2007 STV Counting Rules Committee Condorcet. D. Baumeister, S. Dennisen, L. Rey, Winner Determination and Scoring Rules Inspired Rules Manipulation in Minisum and Minimax Committee Elections. ADT 2015 Committee R. Bredereck, P. Faliszewski, A. Kaczmarczyk, R. Niedermeier, P. Elections Skowron, N. Talmon: Robustness Among Multiwinner Voting P. Faliszewski, P. Skowron, N. Talmon, Bribery as a Measure of Candidate Success: Complexity Results for Approval-Based Multiwinner Rules, AAMAS 2017 robustness Shift-bribery R. Bredereck, P. Faliszewski, R. Niedermeier, N. Talmon: Complexity of Shift Bribery in Committee Elections. AAAI 2016 Rules. SAGT 2017 SEO control ns latio Bribery ipu man Success measure Changing Election Results

Summary • Committee elections – Many computational problems – Parameters • #candidates – easy for simple rules, but maybe more complicated for iterative rules (and result-changing problems) • #voters – partial results here and there (many missing) • #committee size – some W[2]-hardness…. FPT-AS – Problems • Winner determination: Still work to be done (Condorcet!) • Bribery / control / manipulation only preliminary results – Do something with committee size!

Thank you!

SNTV Bloc k-Borda CC STV PAV